Consider the following model of the IS curve without an international sector (i.e., imports = exports = 0). Final expenditures are characterized by the following three equations:

Consumption:
𝐶
𝑡
𝑌
¯
𝑡
=
𝑎
¯
𝑐
+
𝑏
¯
𝑐
(
𝑅
𝑡
−
𝑟
¯
)
Investment:
𝐼
𝑡
𝑌
¯
𝑡
=
𝑎
¯
𝑖
−
𝑏
¯
𝑖
(
𝑅
𝑡
−
𝑟
¯
)
Government expenditure:
𝐺
𝑡
=
𝑎
¯
𝑔
𝑌
¯
𝑡

Assume that
∞

𝑏
¯
𝑖

𝑏
¯
𝑐

0
.

In contrast to the assumptions we made in class, consumption in this version of the model responds to changes in the real interest rate.

Based on this information, the IS curve is:

Question

Consider the following model of the IS curve without an international sector (i.e., imports = exports = 0). Final expenditures are characterized by the following three equations:

Consumption: 
𝐶
𝑡
𝑌
¯
𝑡
=
𝑎
¯
𝑐
+
𝑏
¯
𝑐
(
𝑅
𝑡
−
𝑟
¯
)
Investment: 
𝐼
𝑡
𝑌
¯
𝑡
=
𝑎
¯
𝑖
−
𝑏
¯
𝑖
(
𝑅
𝑡
−
𝑟
¯
)
Government expenditure: 
𝐺
𝑡
=
𝑎
¯
𝑔
𝑌
¯
𝑡

Assume that 
∞
>
𝑏
¯
𝑖
>
𝑏
¯
𝑐
>
0
.

In contrast to the assumptions we made in class, consumption in this version of the model responds to changes in the real interest rate.

Based on this information, the IS curve is:

BlackTom AI · Accepted Answer

Question restatement: We’re looking at a closed-economy IS curve model (no international sector) with three expenditure equations: C_t/Ybar_t = a_bar_c + b_bar_c (R_t − r_bar), I_t/Ybar_t = a_bar_i − b_bar_i (R_t − r_bar), and G_t = a_bar_g Ybar_t. Given assumptions ∞ > b_bar_i > b_bar_c > 0 and that consumption now responds to changes in the real interest rate, we are asked to determine the IS curve’s slope.

Option A: horizontal. This would imply that output Y is completely unresponsive to changes in the real interest rate R_t. In this model, both consumption and investment respond to R_t (via the terms involving (R_t − r_bar)), so achange in R_t changes C and I, and hence Y. Therefore, a purely horizontal IS is inconsistent with the stated equations and the stated responsiveness. This option mischaracterizes the mechanism by which R_t affects aggregate demand.

Option B: vertical. A vertical IS would imply that Y is fixed regardless of R_t, which would require autonomous spending to completely insulate Y from R_t movements or an infinite sensitivity that cancels out in equilibrium. Here, Y responds through C and I to R_t with finite, positive coefficients b_bar_c and b_bar_i, so there is a nonzero slope. This option therefore misstates the actual linkage implied by the given equations.

Option C: less steeply sloped than the “standard” IS curve with b_bar_c = 0. If b_bar_c were zero, only investment responds to R_t, via −b_bar_i (R_t − r_bar). In the present setup, consumption also responds to R_t through −b_bar_c (R_t − r_bar). Since both C and I fall when R_t increases (given the negative sensitivity to R_t − r_bar), total changes in Y due to a given change in R_t would incorporate both channels. Whether the resulting IS curve is steeper or less steep than the bc = 0 benchmark depends on the relative magnitudes of b_bar_i and b_bar_c. Here we are told ∞ > b_bar_i > b_bar_c > 0, i.e., consumption’s responsiveness to R_t is positive but smaller than investment’s. The combined slope contribution is −(b_bar_i + b_bar_c) times the change in R_t, which in this setup would generally be larger in magnitude than the −b_bar_i term alone. That would suggest a steeper (not less steep) IS curve relative to the bc = 0 case. However, the option posits it is less steep. The correct interpretation requires comparing the actual marginal effect on Y of a unit change in R_t between the two specifications: with bc = 0, ΔY ≈ −b_bar_i ΔR_t; with bc > 0, ΔY ≈ −(b_bar_i + b_bar_c) ΔR_t. Because b_bar_c > 0, the latter magnitude is larger, implying a steeper slope. Therefore, this option contradicts the model’s mechanics as described, unless there is an unconventional normalization or a different way the standard is defined in the course. The statement as written is therefore at odds with the straightforward aggregation of the two reaction channels.

Option D: Not enough information is given. The problem provides explicit functional forms and parameter inequalities for b_bar_i and b_bar_c, and states that consumption responds to R_t. Given these, one can in principle compute the IS slope (up to the usual multiplier normalization). Since the model specifies how C and I depend on R_t, and G is exogenous, there is sufficient information to judge how Y responds to R_t. Thus, claiming not enough information would be incorrect. This option ignores the explicit structure provided for C, I, and G and the relative sizes of the coefficients.

Summary note: Across the options, the intuition is that when both consumption and investment respond to the real interest rate, the IS curve generally reflects the combined effect of both channels. The sign and magnitude of the slope follow from dY/dR_t = −(b_bar_c + b_bar_i) given a simplified linear aggregation, which tends to make the curve steeper (more negative) than the case where only investment responds (b_bar_c = 0). The options present competing characterizations of the slope, and each hinges on how one interprets the comparative statics under bc > 0 versus bc = 0. The key takeaway is that C’s responsiveness adds to the slope of the IS curve beyond the investment channel, altering its steepness relative to the bc = 0 benchmark; careful reading is required to align with the precise articulation of “steepness” used in the course materials.}

FA25 ECON 302 002 Homework #8 (IS Curve)

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